Some explicit formulas for the Brownian bridge,
Brownian meander and Bessel process under uniform
sampling
Mathieu Rosenbaum, Marc Yor
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Mathieu Rosenbaum, Marc Yor. Some explicit formulas for the Brownian bridge, Brownian
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Some explicit formulas for the Brownian bridge, Brownian
meander and Bessel process under uniform sampling
Mathieu Rosenbaum and Marc Yor
LPMA, University Pierre et Marie Curie (Paris 6)
November 8, 2013
Abstract
We show that simple explicit formulas can be obtained for several relevant quantities
related to the laws of the uniformly sampled Brownian bridge, Brownian meander and
three dimensional Bessel process. To prove such results, we use the distribution of a triplet
of random variables associated to the pseudo-Brownian bridge given in [8], together with
various relationships
between the laws of these four processes. Finally, we consider the
√
variable BUT1 / T1 , where B is a Brownian motion, T1 its first hitting time of level one
and U a uniform random variable independent of B. This variable is shown to be centered
in [3, 8]. The results obtained here enable us to revisit this intriguing property through
an enlargement of filtration formula.
Keywords: Brownian motion, Brownian bridge, Brownian meander, pseudo-Brownian bridge,
Bessel process, uniform sampling, local times, hitting times, enlargement of filtration.
1
Introduction
Let (Bt , t ≥ 0) be a standard Brownian motion, T1 its first hitting time of level one, and U a
uniform random variable on [0, 1], independent of B. In [3], it is first shown that the random
variable α defined by
BU T
(1)
α= √ 1
T1
is centered. Intrigued by this property, we determined the distribution of this variable, which
is expressed in [8] under the following form, where = denotes equality in law:
L
1
α = ΛL1 − |B1 |,
L
2
(2)
with Lt the local time at point 0 of B at time t and Λ a uniform random variable on [0, 1],
independent of (|B1 |, L1 ). The centering property is easily recovered from (2) since
1
1
E[ΛL1 − |B1 |] = E[L1 − |B1 |] = 0.
2
2
In fact, in [8], a preliminary to the proof of (2) is to obtain the law of a triplet of random variables defined in terms of the pseudo-Brownian bridge introduced in [1], see Section 2 below.
1
In this paper, we show that the law of this triplet enables us to derive several unexpected
simple formulas for various quantities related to some very classical Brownian type processes,
namely the Brownian bridge, the Brownian meander and the three dimensional Bessel process. More precisely, we focus on distributional properties of these processes when sampled
with an independent uniform random variable. Thus, this work can be viewed as a modest
complement to the seminal paper by Pitman, see [6], where the laws of these processes when
sampled with (several) independent uniform random variables are already studied.
The paper is organized as follows. In Section 2, we give some preliminary results related
to the law of (|B1 |, L1 ). Indeed, they play an important role in the proofs. Distributional
properties for the Brownian bridge are established in Section 3 whereas the Brownian meander
and the three dimensional Bessel process are investigated in Section 4. Finally, in Section
5, we reinterpret the fact that α is centered through the lenses of an enlargement formula
for the Brownian motion with the time T1 due to Jeulin, see [5]. In particular we show that
this centering property can be translated in terms of the expectation of the random variable
U/(RU R12 ), where R is a three dimensional Bessel process and U a uniform random variable
on [0, 1] independent of R.
2
Some preliminary results about the law of (|B1|, L1)
Before dealing with the Brownian bridge, the Brownian meander and the three dimensional
Bessel process, we give here some preliminary results related to the distribution of the couple
(|B1 |, L1 ). These results will play an important role in the proofs of our main theorems.
It is well known that the law of (B1 , L1 ) admits a density on R × R+ . Its value at point (x, l)
is given by
(l + |x|)2 1
√ (|x| + l)exp −
.
(3)
2
2π
For l ≥ 0, we set
Z
H(l) = el
+∞
2 /2
dxe−x
2 /2
.
l
The following consequences of (3) shall be useful in the sequel.
Proposition 2.1. Let l ≥ 0. We have the double identity
E[L1 ||B1 | = l] = E[|B1 ||L1 = l] = H(l).
(4)
Furthermore, one has
1
],
N 2 + l2
where N denotes a standard Gaussian random variable.
H(l) = lE[
(5)
Proof. We start with the proof of (4). Of course, it can be deduced from (3) at the cost of
some integrations. We prefer the following arguments. First, the equality on the left hand
side of (4) stems from the symmetry of the law of (|B1 |, L1 ), which is obvious from (3). Thus,
we now have to show the second equality in (4). This easily follows from the identity
Z L1
dxφ(x)],
(6)
E[φ(L1 )|B1 |] = E[
0
2
which is valid for any bounded measurable function φ. Indeed, assuming (6) for a moment,
using the fact that
L1 = |B1 |,
L
we may write (6) as
Z
0
+∞
dle
−l2 /2
φ(l)E[|B1 ||L1 = l] =
+∞
Z
dlφ(l)
0
Z
+∞
dxe−x
2 /2
.
l
Hence, since this is true for every bounded measurable function φ, we get
Z +∞
2
−l2 /2
dxe−x /2 ,
e
E[|B1 ||L1 = l] =
l
which is the desired result for (4).
It remains to prove (6) for a generic bounded measurable function φ. Remark that the formula
φ(Lt )|Bt | =
Z
t
dBs φ(Ls )sign(Bs ) +
Z
t
dLs φ(Ls )
0
0
is a very particular case of the balayage formula, see [7], page 261. It now suffices to take
expectation on both sides of this last equality to obtain (6).
We now give the proof of the second part of Proposition 2.1. First, note that
Z +∞
1
2
2
E[ 2
dve−vl E[e−vN ].
]
=
N + l2
0
Using the Laplace transform of N 2 , we obtain
E[
1
]=
2
N + l2
Z
+∞
0
2
e−vl /2
dv √
.
2 1+v
(7)
Then remark that thanks to the change of variable x2 = (1 + v)l2 , we get
Z +∞
Z +∞
2
l
l2 /2
−x2 /2
e−vl /2 .
H(l) = e
dxe
=
dv √
2 1+v
l
0
This together with (7) gives the second part of Proposition 2.1.
3
The Brownian bridge under uniform sampling
Before giving our theorem on the uniformly sampled Brownian bridge, we recall a result
related to the pseudo-Brownian bridge established in [8], and which is the key to most of the
proofs in this paper. The pseudo-Brownian bridge was introduced in [1] and is defined by
Buτ
( √ 1 , u ≤ 1),
τ1
3
with (τl , l > 0) the inverse local time process:
τl = inf{t, Lt > l}.
This pseudo-Brownian bridge is equal to 0 at time 0 and time 1 and has the same quadratic
variation as the Brownian motion. Thus, it shares some similarities with the Brownian bridge,
which explains its name. Let U be a uniform random variable on [0, 1] independent of B.
The following theorem is proved in [8].
Theorem 3.1. There is the identity in law
1
BU τ
1
( √ 1 , √ , LU τ1 ) = ( B1 , L1 , Λ),
L 2
τ1
τ1
with Λ a uniform random variable on [0, 1], independent of (B1 , L1 ).
In other words, LU τ1 is a uniform random variable on [0, 1], independent of the pair
1
BU τ
( √ 1 , √ ),
τ1
τ1
which is distributed as ( 21 B1 , L1 ).
To deduce some properties of the Brownian bridge from Theorem 3.1, the idea is to use an
absolute continuity relationship between the law of the pseudo-Brownian bridge and that of
the Brownian bridge shown by Biane, Le Gall and Yor in [1]. More precisely, for F a non
negative measurable function on C([0, 1], R), we have
r
Buτ1
1
2 E F √ , u≤1 =
E F b(u), u ≤ 1 0 ,
(8)
τ1
π
λ1
where b(u), u ≤ 1 denotes the Brownian bridge and (λxu , u ≤ 1, x ∈ R) its family of local
times. Let U be again a uniform random variable on [0, 1] independent of b. The following
theorem is easily deduced from Theorem 3.1 together with Equation 8.
Theorem 3.2. For any non negative measurable functions f and g, we have
r
λ0U π 1
0
E f ( B1 , L1 )L1 E[g(Λ)],
E f b(U ), λ1 g 0 =
2
2
λ1
(9)
with Λ a uniform random variable on [0, 1], independent of (B1 , L1 ).
Thus, λ0U /λ01 is a uniform random variable on [0, 1], independent of the pair b(U ), λ01 which
is distributed according to (9) with g = 1.
The following corollary of Theorem 3.2 provides some surprisingly simple expressions for some
densities and (conditional) expectations of quantities related to the Brownian bridge.
Corollary 3.1. The following properties hold:
4
• The variable λ01 admits a density on R+ . Its value at point l ≥ 0 is given by
lexp(−l2 /2).
Hence, λ01 has the same law as
is Rayleigh distributed.
√
2E, with E an exponential random variable. Therefore, λ01
• The density of b(U ) at point y given λ01 = l is given by
E[λy1 |λ01 = l] = (2|y| + l)exp − (2y 2 + 2|y|l) .
Consequently, there is the formula
E
λy1 λ01
= exp(−2y 2 ).
• The density of b(U ) at point y is given by
Z +∞
dzexp(−z 2 /2).
E[λy1 ] =
2|y|
√
Thus, we have b(U ) = 2E(V /2), with E an exponential variable independent of V which is
L
uniformly distributed on [−1, 1].
The first part of Corollary 3.1 is obviously deduced from Theorem 3.2 and is in fact a very
classical result, see [1, 2, 4, 7]. We now prove the second part.
Proof. Let f be a non negative measurable function. First note that
Z
Z
0
1
0
dyf (y)E[λy1 |λ01 = l].
E f b(U ) |λ1 = l = E
duf b(u) |λ1 = l =
0
R
Hence the density of b(U ) at point y given
λ01
= l is equal to
E[λy1 |λ01 = l].
Now, let h denote the density of the couple (B1 , L1 ) given in Equation (3). From Theorem
3.2, we easily get that the density of b(U ) at point y given λ01 = l is equal to
r
√
π lh(2y, l)
= 2πh(2y, l)exp(l2 /2).
2
2
2 lexp(−l /2)
The first statement in the second part of Corollary 3.1 readily follows from Equation (3). For
the second statement, we use the fact that
Z
√ Z +∞
+∞ 1
λy1 y 0
2
dlh(2|y|, l).
dl E[λ1 |λ1 = l] lexp(−l /2) = 2π
E 0 =E
l
λ1
0
0
Using the definition of h, this last expression is equal to
exp(−2y 2 ).
The last identity in Corollary 3.1 is easily deduced from Theorem 3.2 together with Proposition
2.1. Note that this formula can also be found in [9], page 400.
5
4
The Brownian meander and the three dimensional Bessel
process under uniform sampling
In this section, we reinterpret Theorem 3.1 in terms of the Brownian meander and the three
dimensional Bessel process.
4.1
The Brownian meander
We first turn to the translation of Theorem 3.1 in terms of the Brownian meander, denoted
by m(u), u ≤ 1 . To do so, we use an equality in law shown by Biane and Yor in [2]. More
precisely we have
(m(u), iu ), u ≤ 1 = (|b(u)| + λ0u , λ0u ), u ≤ 1 ,
L
where
iu = inf mt .
u≤t≤1
Thus, we can reinterpret Theorem 3.2 as follows.
Theorem 4.1. For any non negative measurable functions f and g, we have
r
iU π 1
E f m(U ), m(1) g
E f ( |B1 | + ΛL1 , L1 )L1 g(Λ) ,
=
m(1)
2
2
with Λ a uniform random variable on [0, 1], independent of (B1 , L1 ).
Let (λ̃y1 , y ≥ 0) denotes the family of local times of m at time 1. Similarly to what we have
done for the Brownian bridge, we are able to retrieve from Theorem 4.1 simple expressions
for the laws of m(1) and m(U ). We state these results in the following corollary.
Corollary 4.1. The following properties hold:
• The variable m(1) is Rayleigh distributed.
• The density of m(U ) at point y ≥ 0 is given by
Z 2y
y
exp(−z 2 /2)dz.
E[λ̃1 ] = 2
y
Thus, we have m(U ) =
L
√
2EW , with E an exponential variable independent of W which is
uniformly distributed on [1/2, 1].
Proof. The proof of the first part of Corollary 4.1 is obvious from Theorem 4.1. We now
consider the second part. Let f be a non negative measurable function. Using Theorem 4.1
together with Equation (3), we get that
Z +∞ Z +∞
x
2
dll(x + l)e−(x+l) /2 E f ( + Λl) .
dx
E f m(U ) =
2
0
0
Now remark that
1
x
E f ( + Λl) =
2
l
6
Z
x/2+l
dνf (ν).
x/2
Therefore, by Fubini’s theorem, we get
Z +∞ Z
dx
E f m(U ) =
0
=
Z
dνf (ν)
+∞
2 /2
dl(x + l)e−(x+l)
0
x/2+l
dνf (ν)
x/2
Z
2ν
dx
0
Z
+∞
2 /2
dl(x + l)e−(x+l)
.
ν−x/2
Thus, the density of m(U ) at point ν is given by
Z
Z +∞
Z 2ν
−(x+l)2 /2
dl(x + l)e
=
dx
0
Z
ν−x/2
=2
2ν
dxexp −
0
Z
2ν
dze−z
2 /2
(x/2 + ν)2 2
.
ν
This ends the proof.
In fact, as it is the case for the Brownian bridge, we can give explicit formulas for several
other quantities related to the Brownian meander, for example the law of m(U ) given m(1).
However, these expressions are not so simple and therefore probably less interesting than
those obtained for the Brownian bridge.
4.2
The three dimensional Bessel process
Finally, let (Rt , t ≥ 0) be a three dimensional Bessel process starting from 0 and
Ju = inf Rt .
u≤t≤1
Using Imhof’s absolute continuity relationship between the law of the meander and that of
the three dimensional Bessel process, see [1, 4], we may rewrite Theorem 4.1 as follows.
Theorem 4.2. For any non negative measurable functions f and g, we have
1
JU E f R(U ), R(1) g
= E f ( |B1 | + ΛL1 , L1 )L21 g(Λ) ,
R(1)
2
with Λ a uniform random variable on [0, 1], independent of (B1 , L1 ).
We finally give the following corollary.
Corollary 4.2. The following properties hold:
• The density of R(1) at point y ≥ 0 is given by
r
2 2
y exp(−y 2 /2).
π
√
• R(U ) = U R(1) and its density at point y ≥ 0 is given by
L
r
2
2
y
π
Z
+∞
exp(−z 2 /2)dz.
y
• The law of R(U ) given R(1) is the same as the law of m(U ) given m(1).
The first two parts of Corollary 4.2 are in fact easily deduced from basic properties of the
three dimensional Bessel process. The last part is a consequence of Imhof’s relation.
7
5
The centering property of α revisited through an enlargement of filtration formula
In this last section, we revisit the centering property of the variable
BU T
α = √ 1,
T1
which is proved in [3] and leads to various developments in [8]. Our goal here is to show that
this result can be recovered from simple properties of the three dimensional Bessel process
sampled at uniform time, together with an enlargement of filtration formula for the Brownian
motion with the time T1 due to Jeulin, see [5].
5.1
Some preliminary remarks on the uniformly sampled Bessel process
Let U be a uniform random variable on [0, 1], independent of the considered Bessel process
R. We start with the two following lemmas on the conditional expectation of the uniformly
sampled Bessel process.
Lemma 5.1. We have
E[RU |R1 = r] =
1
U
|R1 = r] .
r + E[
2
RU
(10)
Consequently,
E
RU R12
Lemma 5.2. We have
E[
1
=
2
r
2
U ] .
+ E[
π
RU R12
U
|R1 = r] = H(r).
RU
Consequently,
E
RU R12
U
= E[
]=
RU R12
r
2
.
π
Remark that we already proved the equality
E
RU R12
=
r
2
π
in [3]. This was in fact the cornerstone of our first proof of the centering property of α. In the
enlargement of filtration approach used here, we will see that instead of RU /R12 , the random
variable U/(RU R12 ) appears naturally.
5.2
Proofs of Lemma 5.1 and Lemma 5.2
We now give the proofs of Lemma 5.1 and Lemma 5.2.
8
5.2.1
Proof of Lemma 5.1
The first part of Lemma 5.1 follows from the identity
Z 1
Rt
dv
E[ |R1 ] = R1 + E[
|R1 ], t ≤ 1,
t
t vRv
(11)
after multiplying both sides by t and integrating in t from 0 to 1. To show (11), we use time
inversion with t = 1/w and
′
Rw
= wR1/w ,
another three dimensional Bessel process. With this notation, using the Ito representation of
the Bessel process, we get
Z w
dt ′
′
′
′
E[Rw − R1 |R1 ] = E[
′ |R1 ],
1 Rt
from which (11) is easily obtained. The second statement in Lemma 5.1 readily follows.
5.2.2
Proof of Lemma 5.2
We start with the proof of the first part of Lemma 5.2. Let
Z 1
v
dv |R1 = r].
ρ = E[
Rv
0
Using the same time inversion trick as for the proof of Lemma 5.1 together with the Markov
property, we get
Z +∞
Z +∞
dt
1
1
dw
′
E[
|R
=
r]
=
E [ ],
ρ=
1
2
′
2 r R′
w
R
(1
+
t)
0
1
w
t
where Pr denotes the law of a Bessel process R′ starting from r. We then use the Doob’s
absolute continuity relationship, that is
Xt∧T0
Pr Ft =
Wr Ft ,
r
where Wr is the Wiener measure associated to a Brownian motion starting at point r, X is
the canonical process and T0 is the first hitting time of 0 by X, see for example [7], Chapter
XI. This together with the fact that
Xt∧T0
= 1T0 >t
Xt
gives
1
1
1
Er [ ′ ] = Wr [T0 > t] = W0 [Tr > t].
Rt
r
r
Therefore,
Z
Tr
dt 1 W0 Tr
1
1
= E [
] = rE[ 2
].
ρ = EW0
2
r
(1 + t)
r
1 + Tr
N + r2
0
Using Equation (5), this is equal to H(r). This ends the proof of the first part of Lemma 5.2.
Using the expression for the density of R1 given in Corollary 4.2, the proof of the second part
readily follows remarking that
r Z
r
r Z
Z +∞
U 2 +∞
2 +∞
2
−x2 /2
−x2 /2
dxe
=
.
dr
dxxe
=
=
E
2
π
π
π
RU R1
r
0
0
9
5.3
An enlargement of filtration approach to the centering property of α
We now revisit the centering property of α through an enlargement of filtration formula. Let
(Ft ) denote the filtration of the Brownian motion (Bt ) and (Ft′ ) the filtration obtained by
initially enlarging (Ft ) with T1 . It is shown in [5] that (Bt ) is a (Ft′ ) semi-martingale. More
precisely,
Z t∧T1
Z t∧T1
1 − Bs
ds
ds
+
,
(12)
Bt = βt −
1 − Bs
T1 − s
0
0
where (βt ) is a (Ft′ ) Brownian motion (in particular it is independent of T1 ). Taking expectation on both sides of (12) at time t = U T1 , we get
Z U T1
Z U T1
1
1
ds 1 − Bs E[α] = −E √
ds
+E √
.
1 − Bs
T1 − s
T1 0
T1 0
Using the change of variable s = uT1 in both integrals, we get
Z U
Z U
1
p
du 1 − BuT1 +E √
du
.
E[α] = −E T1
1−u
T1 0
0 1 − BuT1
Since U is independent of B and uniformly distributed on [0, 1], we get
Z 1
Z 1
p
1
(1 − u) du
E[α] = −E T1
+E √
du(1 − BuT1 ) .
1 − BuT1
T1 0
0
Thus,
2E[α] = −E
p
T1
1
Z
dv
0
1 v
+E √
.
1 − BT1 (1−v)
T1
We now use Williams time reversal result:
T1 , 1 − BT1 (1−v) , v ≤ 1 = γ1 , Rvγ1 , v ≤ 1 ,
L
where
γ1 = sup{s > 0, Rs = 1}.
Hence we obtain
2E[α] = −E
1 V
+E √
,
√
RV γ1 / γ1
T1
with V a uniform random variable on [0, 1], independent of R. From the absolute continuity
relationship between the laws of
√
Rvγ1 / γ1 , v ≤ 1
and (Rv , v ≤ 1), see [1], we get
2E[α] =
r
V
2
− E[
].
π
RV R12
Hence E[α] = 0 if and only if
V
]=
E[
RV R12
r
2
.
π
From Lemma 5.2, the last equality holds. Moreover, it has been shown without the help
of our previous results [3, 8]. Thus, the use of the enlargement formula of [5] provides an
alternative proof of the centering property of α.
10
6
A few words of conclusion
Together with [3] and [8], this paper is our third work where various aspects of the law of
BU T
α= √ 1
T1
are investigated. For example, we have considered its centering property, the explicit form of
its density, which may be directly deduced from Equation (2) and Equation (3), and its Mellin
transform. In the present paper, starting from the pseudo-Brownian bridge, we obtain some
results relative to the Brownian bridge, the Brownian meander and the three dimensional
Bessel process. Imhof type relations between these processes allow to go from one to another.
References
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11